Floating-Point Homomorphic Encryption

Our paper suggests a general method to construct a Floating-Point Homomorphic Encryption (FPHE) scheme that allows the floating-point arithmetics of ciphertexts, thus computing encryptions of most significant bits of m1+m2 and m1m2, given encryptions of floating-point numbers m1 and m2. Our concrete construction of leveled FPHE based on BGV scheme is almost optimal in the sense of noise growth and precision loss. More precisely, given encryptions of d messages with η bits of precision, our scheme of depth dlog de securely computes their product with η−dlog de bits of precision similarly to the case of unencrypted floating-point computation. The required bit size of the largest modulus grows linearly in the depth. We also describe algorithms for evaluating some floating-point arithmetic circuits containing polynomial, multiplicative inverse, and even exponential function, and analyze their complexities and output precisions. With the security parameter λ = 80, our rudimentary implementation takes 315ms and 168ms to compute a product of 16 ciphertexts and a multiplicative inverse of a ciphertext, respectively, when given ciphertexts have 20 bits of precision.